That's exactly what we did last September and didn't have any issues. The best thing to do is hire some type of 4WD like we did just in case as I thought the same thing. Its unusual to get snowed in or roads closed for very long, but it does happen occasionally. By the way the Queenstown/Rosebery drive is stunning. I wish we had more time to look through that area. The rivers, lake crossings, Mount Lyell copper mine, the Queesntown wilderness railway are all amazing.
So worse case if snow was an issue you could stay a night in Queenstown, I hope that's not the case.
You get all that information when you arrive. No need to book before you get there, but once you arrive you should book to be safe. Enjoy the wombats, they are quite something. They come out in the evening around the lodge to graze. Have fun!